- Email: [email protected]
Love and shear horizontal interface waves in a superlattice deposited on a substrate
361
Surface Science 211/212 (1989) 361-367 North-Holland, Amsterdam
LOVE AND SHEAR HORIZONTAL IN A SUPERLATTICE DEPOSITED E. KHOURDIFI
and B. DJAFARI
INTERFACE WAVES ON A SUBSTRATE
ROUHANI
Loboratoire de Physique du Glide, Fact& des Sciences et Techniques, 4, rue des F&es Lumi.+e, 68093 Mulhouse Cedex, France
lJniuersit6 de Haute Alsace,
Received 14 June 1988; accepted for publication 23 September 1988
The existence of shear horizontal interface vibrations localized at the boundary between a substrate and a semi-infinite superlattice made of alternating layers of two elastic media is investigated theoretically. It is shown that in contrast to the case of an interface between two homogeneous media, such localized waves may appear inside the gaps of the superlattice (resulting from the folding of the Brillouin zone along the superlattice axis) and even below the bottom of the bulk bands. Their existence and dispersion are dependent upon the relative parameters of the superlattice and the substrate, but also upon the nature of the film in the superlattice which is in contact with the substrate. We also investigate the behaviour of the Love waves when a finite superlattice is deposited on a substrate as a function of the superlattice thickness and of the wave vector k ,, parallel to the layers.
The bulk and surface vibrations of superlattices have been extensively studied during the last years [l], both theoretically and experimentally. The Brillouin scattering enables one to observe some of the surface waves (Rayleigh, Sezawa, Love) localized at the surface of a semi-infinite superlattice or of a finite superlattice deposited on a substrate [2-lo]. These measurements have been used to investigate the elastic constants of superlattices and deduce some structural information on these materials, specially near the interfaces. In some of these works the superlattice is considered as an effective homogeneous adlayer on a substrate [7,8,10]. In this paper we investigate the shear horizontal interface and surface vibrations of a superlattice in contact with a substrate. First, we show that in contrast to the case of an interface between two elastic homogeneous media, localized waves may exist at the superlattice-substrate interface; their frequencies belong to the gaps which separate the bulk bands of the superlattice and may even be situated below these bands. Then we consider a finite superlattice deposited on a substrate and study the dispersion of Love waves (which are exponentially decaying into the substrate) and their deviations from the results of the effective medium approximation. The dispersion relations can be obtained by using either a transfer matrix method [11,12] or the 0039-6028/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
E. Khourdrfi, B. D&n-i
362
Rouhan: / Low and shear hori:ontul interface MWJ~S
Table 1 Elastic parameters of the materials Type of layer
c,, (10”
Y DY
Y and Dy c,* (IO”
N/m*)
7.19 7.31
2.85 2.53
interface response theory present a few illustrations In our calculation with (0001)
numerical
examples
lattice
in the sagittal
[12] without
and
k,,
(synthesized
(Pkg/m’)
2.431 2.40
4450 8560
to be of hexagonal
in which case the shear horizontal
from those polarized interfaces
c44 (IO’” N/m2)
[13]. After a brief outline of the theory of the interface and Love waves.
all the media are assumed
interfaces,
be transposed
N/m’)
along
further the high
plane [12]. However calculation symmetry
are given for an isotropic recently
Cd,) and mass densities
[14]) whose
elastic
to cubic
symmetry
waves are decoupled these results can also materials
[IOO] or [110] substrate
we shall
with (001)
directions.
and a Y-Dy
constants
(C,,
C,,
and C&,
(p and p’) are given in table 1; the thicknesses
to be equal. Let x3 be the axis of the superlattice and X, the direction
The superd and
d’ of the layers are assumed
of k,,.
For shear
horizontal waves only the u2 component of the displacement field is different from zero. In the substrate, occupying the negative x3 half-space, an exponentially decaying
wave can be written as
U:“‘( X, I) = ‘4, ea\x, el’x,,^‘- wr), where w is the frequency,
(1)
the index s refers to the substrate
and
C(“)kZ _ P(“)w2 66
2
II
%,
C(S) 44
(2)
.
In the superlattice
the displacement
field in the first type of film belonging
to
the nth unit cell takes the form [12] u:“)(
x,
t)
=
(A(N)
e-a-xY’
+ B(n)
where xin) = xX - (n - 1)D (2). A similar relation with the superlattice. The transfer enable one to relate the introduce a wave vector k, A’“’ (or B’“‘)
eaxY’)
e’(% ‘Ipw’),
(3)
is a local coordinate and a2 is defined as in eq. primed quantities holds in the second medium of matrix method [11,12] and the Floquet’s theorem displacement field in successive unit cells and along x3 such that
= ‘4’0’ (or B(O)) et Ik,!lD,
where D = d + d’ is the period of the superlattice. The wave vector k, is real (respectively complex) if the frequency o belongs to a bulk band (respectively to a gap).
E. Khourdifi, B. Djufari Rouhnni / Low and shear horrrontal interface waues
In searching
interface
waves localized
at the superlattice-substrate
we only keep the term in e * ik+D which is exponentially interface.
On the other hand in a finite superlattice
363
boundary
decaying
far from the
both types of terms (eikqnD
and e PikjnD) should be included in solution (2). Using the interface and the free surface boundary conditions we obtain the following dispersion relations: (i) For the interface
waves, when medium
1 is in contact
with the substrate:
e”“(l+F,)(F-F~‘)S;-(1-F,){e~“d[2C;-(F+F-’)S;]
-eik+}
=O, (5)
and C, =cosh(&),
C,‘=cosh(a’d’),
S, =sinh(old),
S;=sinh(a’d’). (6b)
(ii) For the Love waves, when medium
1 is both at the interface
and at the
free surface: - 2 cos( k,ND) x {(F(iii)
sin( k,D)(
F-‘)F$S;
C, + F,S,)
- [2&C,‘+
+ sin( k,ND)
(F+
Fp’)C,S;](S,
For the Love waves, when medium
+ FIG,)}
= 0.
1 is at the interface
(7a)
and medium
2
at the surface: 2 sin( k,D)
cos( k,Nn)
+ sin( k,ND)
x{(F~‘-F)S,S,‘+2F,(S,C,‘+F~‘C,S;)} Three
other
interface
relations
and Love
=O.
can be obtained waves
0)
by interchanging
are thus dependent
upon
media
which is in contact with the substrate or at the free surface. A few illustrations of these results will now be discussed. interface substrate. substrate =JW
waves for a Y-Dy These
localized
parameters remains
(C$),
superlattice modes
are
Though
of the film
Fig. 1 gives the
when a Dy layer is in contact represented
p(‘)) so that the substrate
constant.
1 and 2; the
the nature
for
several
velocity
this representation
sets
with the of
of sound
was chosen
the C/‘)
for the
sake of clarity, it should be pointed out that the dispersion curves are rather sensitive to the value of Cz’ while they are not much affected by the variation of p(‘) alone. This is due to the dependence of the normal stress boundary condition upon the ratio y = C@/C,. In the limit y -+ 0 eq. (5) gives the surface modes of a semi-infinite superlattice in contact with the vacuum (let us recall from ref. [12] that when a Dy (respectively Y) layer is at the free surface, one surface branch exists below the bottom of the bulk bands (respectively higher surface extreme limit
branches appear in the gaps between the bands)). In the other y -+ cc the amplitude of the vibrations goes to zero at the
364
E. Khourdifi, B. DJaJai-iRouhani / Love and shear horizontal interface waues
lo-
Fig. 1. Dispersion of shear horizontal interface modes when the substrate is in contact with a Dy film in the superlattice, for different sets of substrate parameters. The velocity of sound in the substrate is taken to be constant ( CrrS)= 2C,(Dy)) and indicated by the heavy line. The dispersion curves are given for the ratio y = C.&‘/C,(Dy) equal to cc, (---); 7 (-. -. -. -), 3 (- . -), 0.3 (,.. . ..)andO(corresponding to the free surface modes). The branches corresponding to y = 30 are continued inside the bulk bands of the substrate because they are independent of the substrate velocity of sound. The shaded areas are the bulk bands of the superlattice. Dimensionless quantities are reported on both axes.
interface and remains vanishingly small in the substrate. When y decreases from cc (fig. 1) the frequencies of the interface waves decrease until the corresponding branches merge into the bulk bands of the superlattice. For lower values of y, an interface branch is even extracted below the bottom of the bulk bands merging into these bands at a finite value of k ,, D. When the substrate is in contact with a Y film in the superlattice, only interface waves inside the gaps between the bulk bands may exist; they originate from surface waves [12] for relatively small values of Cg’ and disappear into the bands when C$ increases. Fig. 2 represents the Love modes of a finite superlattice deposited on a substrate assuming that Dy layers are both at the interface and at the free surface. For the sake of clarity the superlattice only contains N = 5 layers of Y and N + 1 = 6 layers of Dy. The interface and free surface waves discussed above can be distinguished in fig. 2, even for such a small number of periods N. The other branches in fig. 2 correspond to propagating waves in the superlattice; their number increases with N, leading to the bulk bands of the infinite superlattice in the limit N -+ cc.
E. Khourdifi, B. Djafari Rouhani / Love and shear horizontal interface waues
365
Fig. 2. Dispersion of Love waves for a finite superlattice with Dy layers at both its ends and The labels i and s N = 5, deposited on a substrate with C,(‘) = 2C,(Dy) and C$’ = 5C,(Dy). respectively refer to the interface and free surface modes discussed in fig. 1.
The case of a superlattice with Y at the free surface and Dy at the interface provides the possibility of having one surface and one interface branch in the same gap; these modes may interact together especially when the number of periods N is small. For values of k ,,D 5 1, a comparison of the Love waves velocities using either the exact calculations (eqs. (7)) or the effective medium approximation (EMA) is given in fig. 3 as a function of k,, L, where L is the thickness of the superlattice. The curves 1 (respectively 2) correspond to a superlattice with Dy (respectively Y) layers at both ends and containing 5.5 periods, i.e. N = 5. We also obtain results very close to curves 1 (respectively 2) when the medium at the surface is still Dy (respectively Y) but the film at the interface is Y (respectively Dy). In fig. 3, at k,, L = 1 for example, the velocity difference between curves 1 and 2 is of the order of 50 ms-‘. The EMA results (dashed curves in fig. 3) are obtained by replacing the superlattice with an homogeneous material whose parameters p”, Ct6 and l/C& are given by the arithmetic averages of the corresponding quantities in the superlattice; these curves are of course independent of the number of periods N and the nature of the utmost layers in the superlattice. The exact results become closer to the EMA curves when N increases; this also shows that for a given thickness L of the superlattice one obtains different Love waves velocities by varying simultaneously N and D.
366
E. Khourdifi,
B. Djafari Rouhanr
OS_ 1
/ Love and shear horizontal
1~ 5
I
4
6
mterface
waues
kfA
Fig. 3. Velocities of Love waves as a function of k ,, L, where L is the thickness of the superlattice. The curves 1 (respectively 2) correspond to a superlattice with N = 5 and Dy (respectively Y) layers at its ends. The substrate is the same as in fig. 2. The dashed curves are the result of the effective medium approximation.
In conclusion we have shown for the first time the possibility of shear horizontal interface waves at the boundary between a superlattice and a substrate. It is worth to mention that such a mode does not exist if the substrate is one of the two materials out of which the superlattice is fabricated. For a finite superlattice we have discussed the dispersion of the Love waves as a function of k ,,D and the number of periods N. These results are dependent upon the nature of the interface and surface layers in the superlattice.
References PI See for example:
J. Warlock, in: Proc. 2nd Intern. Conf. on Phonon Physics (World Scientific, Singapore, 1985) p. 506; M.V. Klein, IEEE J. Quantum Electron. 22 (1986) 1760; B. Djafari Rouhani, Lecture Notes in Physics, Vol. 285 (Springer, Berlin, 1987) p, 148. PI A. Kueny, M. Grimsditch, K. Miyano, I. Banerjee. C. Falco and I.K. Schuller, Phys. Rev. Letters 48 (1982) 166. R. Vacher, J. Kervarec and A. Regreny, Phya. Rev. B [31 J. Sapriel, J.C. Michel, J.C. Toledano. 28 (1983) 2007. A. Kueny, C.M.Falco and 1.K. Schuller. 141 M.R. Kan. C.S.L. Chun, G. Felcher. M. Grimsditch. Phys. Rev. B 27 (1983) 718. f51 R. Danner, R.P. Huebener, C.S.L. Chun. M. Grimsditch and I.K. Schuller, Phys. Rev. B 33 (1986) 3696.
E. Khourdifi, 8. Djafari Rouhani / Low and shear honzontal interface waues
367
[6] P. Bisanti, M.B. Brodsky, G.P. Felcher, M. Grimsditch and L. Sill, Phys. Rev. B 35 (1987) 7813. [7] J.A. Bell, W.R. Bennett, R. Zanoni, G.I. Stegeman, CM. Falco and F. Nizzoli, Phys. Rev. B 35 (1987) 4127. [S] P. Baumgart, B. Hillebrands, R. Mock, G. Giintherodt, A. Boufelfel and CM. Falco, Phys. Rev. B 34 (1986) 9004. 191 J.A. Bell, W.R. Bennett, R. Zanoni, G.I. Stegeman, CM. Falco and CT. Seaton, Solid State Commun. 64 (1987) 1339. [IO] J.A. Bell, R.J. Zanoni, CT. Seaton, G.I. Stegeman, W.R. Bennett and CM. Falco, Appl. Phys. Letters 9 (1987) 652. [ll] B. Djafari Rouhani, L. Dobrzynski, 0. Hardouin Duparc, R.E. Camely and A.A. Maradudin, Phys. Rev. B 28 (1983) 1711; R.E. Camley, B. Djafari Rouhani, L. Dobrzynski and A.A. Maradudin, Phys. Rev. B 27 (1983) 7318. (121 A. Nougaoui and B. Djafari Rouhani, Surface Sci. IS5 (1987) 125. [13] L. Dobrzynski, Surface Sci. 182 (1987) 362. [14] M.B. Salamon. S. Sinha, J.J. Rhyne, J.E. Cunningham R.W. Erwin, J. Borchers and C.P. Flynn, Phys. Rev. Letters 56 (1986) 259.
Surface Science 211/212 (1989) 361-367 North-Holland, Amsterdam
LOVE AND SHEAR HORIZONTAL IN A SUPERLATTICE DEPOSITED E. KHOURDIFI
and B. DJAFARI
INTERFACE WAVES ON A SUBSTRATE
ROUHANI
Loboratoire de Physique du Glide, Fact& des Sciences et Techniques, 4, rue des F&es Lumi.+e, 68093 Mulhouse Cedex, France
lJniuersit6 de Haute Alsace,
Received 14 June 1988; accepted for publication 23 September 1988
The existence of shear horizontal interface vibrations localized at the boundary between a substrate and a semi-infinite superlattice made of alternating layers of two elastic media is investigated theoretically. It is shown that in contrast to the case of an interface between two homogeneous media, such localized waves may appear inside the gaps of the superlattice (resulting from the folding of the Brillouin zone along the superlattice axis) and even below the bottom of the bulk bands. Their existence and dispersion are dependent upon the relative parameters of the superlattice and the substrate, but also upon the nature of the film in the superlattice which is in contact with the substrate. We also investigate the behaviour of the Love waves when a finite superlattice is deposited on a substrate as a function of the superlattice thickness and of the wave vector k ,, parallel to the layers.
The bulk and surface vibrations of superlattices have been extensively studied during the last years [l], both theoretically and experimentally. The Brillouin scattering enables one to observe some of the surface waves (Rayleigh, Sezawa, Love) localized at the surface of a semi-infinite superlattice or of a finite superlattice deposited on a substrate [2-lo]. These measurements have been used to investigate the elastic constants of superlattices and deduce some structural information on these materials, specially near the interfaces. In some of these works the superlattice is considered as an effective homogeneous adlayer on a substrate [7,8,10]. In this paper we investigate the shear horizontal interface and surface vibrations of a superlattice in contact with a substrate. First, we show that in contrast to the case of an interface between two elastic homogeneous media, localized waves may exist at the superlattice-substrate interface; their frequencies belong to the gaps which separate the bulk bands of the superlattice and may even be situated below these bands. Then we consider a finite superlattice deposited on a substrate and study the dispersion of Love waves (which are exponentially decaying into the substrate) and their deviations from the results of the effective medium approximation. The dispersion relations can be obtained by using either a transfer matrix method [11,12] or the 0039-6028/89/$03.50 0 Elsevier Science Publishers B.V. (North-Holland Physics Publishing Division)
E. Khourdrfi, B. D&n-i
362
Rouhan: / Low and shear hori:ontul interface MWJ~S
Table 1 Elastic parameters of the materials Type of layer
c,, (10”
Y DY
Y and Dy c,* (IO”
N/m*)
7.19 7.31
2.85 2.53
interface response theory present a few illustrations In our calculation with (0001)
numerical
examples
lattice
in the sagittal
[12] without
and
k,,
(synthesized
(Pkg/m’)
2.431 2.40
4450 8560
to be of hexagonal
in which case the shear horizontal
from those polarized interfaces
c44 (IO’” N/m2)
[13]. After a brief outline of the theory of the interface and Love waves.
all the media are assumed
interfaces,
be transposed
N/m’)
along
further the high
plane [12]. However calculation symmetry
are given for an isotropic recently
Cd,) and mass densities
[14]) whose
elastic
to cubic
symmetry
waves are decoupled these results can also materials
[IOO] or [110] substrate
we shall
with (001)
directions.
and a Y-Dy
constants
(C,,
C,,
and C&,
(p and p’) are given in table 1; the thicknesses
to be equal. Let x3 be the axis of the superlattice and X, the direction
The superd and
d’ of the layers are assumed
of k,,.
For shear
horizontal waves only the u2 component of the displacement field is different from zero. In the substrate, occupying the negative x3 half-space, an exponentially decaying
wave can be written as
U:“‘( X, I) = ‘4, ea\x, el’x,,^‘- wr), where w is the frequency,
(1)
the index s refers to the substrate
and
C(“)kZ _ P(“)w2 66
2
II
%,
C(S) 44
(2)
.
In the superlattice
the displacement
field in the first type of film belonging
to
the nth unit cell takes the form [12] u:“)(
x,
t)
=
(A(N)
e-a-xY’
+ B(n)
where xin) = xX - (n - 1)D (2). A similar relation with the superlattice. The transfer enable one to relate the introduce a wave vector k, A’“’ (or B’“‘)
eaxY’)
e’(% ‘Ipw’),
(3)
is a local coordinate and a2 is defined as in eq. primed quantities holds in the second medium of matrix method [11,12] and the Floquet’s theorem displacement field in successive unit cells and along x3 such that
= ‘4’0’ (or B(O)) et Ik,!lD,
where D = d + d’ is the period of the superlattice. The wave vector k, is real (respectively complex) if the frequency o belongs to a bulk band (respectively to a gap).
E. Khourdifi, B. Djufari Rouhnni / Low and shear horrrontal interface waues
In searching
interface
waves localized
at the superlattice-substrate
we only keep the term in e * ik+D which is exponentially interface.
On the other hand in a finite superlattice
363
boundary
decaying
far from the
both types of terms (eikqnD
and e PikjnD) should be included in solution (2). Using the interface and the free surface boundary conditions we obtain the following dispersion relations: (i) For the interface
waves, when medium
1 is in contact
with the substrate:
e”“(l+F,)(F-F~‘)S;-(1-F,){e~“d[2C;-(F+F-’)S;]
-eik+}
=O, (5)
and C, =cosh(&),
C,‘=cosh(a’d’),
S, =sinh(old),
S;=sinh(a’d’). (6b)
(ii) For the Love waves, when medium
1 is both at the interface
and at the
free surface: - 2 cos( k,ND) x {(F(iii)
sin( k,D)(
F-‘)F$S;
C, + F,S,)
- [2&C,‘+
+ sin( k,ND)
(F+
Fp’)C,S;](S,
For the Love waves, when medium
+ FIG,)}
= 0.
1 is at the interface
(7a)
and medium
2
at the surface: 2 sin( k,D)
cos( k,Nn)
+ sin( k,ND)
x{(F~‘-F)S,S,‘+2F,(S,C,‘+F~‘C,S;)} Three
other
interface
relations
and Love
=O.
can be obtained waves
0)
by interchanging
are thus dependent
upon
media
which is in contact with the substrate or at the free surface. A few illustrations of these results will now be discussed. interface substrate. substrate =JW
waves for a Y-Dy These
localized
parameters remains
(C$),
superlattice modes
are
Though
of the film
Fig. 1 gives the
when a Dy layer is in contact represented
p(‘)) so that the substrate
constant.
1 and 2; the
the nature
for
several
velocity
this representation
sets
with the of
of sound
was chosen
the C/‘)
for the
sake of clarity, it should be pointed out that the dispersion curves are rather sensitive to the value of Cz’ while they are not much affected by the variation of p(‘) alone. This is due to the dependence of the normal stress boundary condition upon the ratio y = C@/C,. In the limit y -+ 0 eq. (5) gives the surface modes of a semi-infinite superlattice in contact with the vacuum (let us recall from ref. [12] that when a Dy (respectively Y) layer is at the free surface, one surface branch exists below the bottom of the bulk bands (respectively higher surface extreme limit
branches appear in the gaps between the bands)). In the other y -+ cc the amplitude of the vibrations goes to zero at the
364
E. Khourdifi, B. DJaJai-iRouhani / Love and shear horizontal interface waues
lo-
Fig. 1. Dispersion of shear horizontal interface modes when the substrate is in contact with a Dy film in the superlattice, for different sets of substrate parameters. The velocity of sound in the substrate is taken to be constant ( CrrS)= 2C,(Dy)) and indicated by the heavy line. The dispersion curves are given for the ratio y = C.&‘/C,(Dy) equal to cc, (---); 7 (-. -. -. -), 3 (- . -), 0.3 (,.. . ..)andO(corresponding to the free surface modes). The branches corresponding to y = 30 are continued inside the bulk bands of the substrate because they are independent of the substrate velocity of sound. The shaded areas are the bulk bands of the superlattice. Dimensionless quantities are reported on both axes.
interface and remains vanishingly small in the substrate. When y decreases from cc (fig. 1) the frequencies of the interface waves decrease until the corresponding branches merge into the bulk bands of the superlattice. For lower values of y, an interface branch is even extracted below the bottom of the bulk bands merging into these bands at a finite value of k ,, D. When the substrate is in contact with a Y film in the superlattice, only interface waves inside the gaps between the bulk bands may exist; they originate from surface waves [12] for relatively small values of Cg’ and disappear into the bands when C$ increases. Fig. 2 represents the Love modes of a finite superlattice deposited on a substrate assuming that Dy layers are both at the interface and at the free surface. For the sake of clarity the superlattice only contains N = 5 layers of Y and N + 1 = 6 layers of Dy. The interface and free surface waves discussed above can be distinguished in fig. 2, even for such a small number of periods N. The other branches in fig. 2 correspond to propagating waves in the superlattice; their number increases with N, leading to the bulk bands of the infinite superlattice in the limit N -+ cc.
E. Khourdifi, B. Djafari Rouhani / Love and shear horizontal interface waues
365
Fig. 2. Dispersion of Love waves for a finite superlattice with Dy layers at both its ends and The labels i and s N = 5, deposited on a substrate with C,(‘) = 2C,(Dy) and C$’ = 5C,(Dy). respectively refer to the interface and free surface modes discussed in fig. 1.
The case of a superlattice with Y at the free surface and Dy at the interface provides the possibility of having one surface and one interface branch in the same gap; these modes may interact together especially when the number of periods N is small. For values of k ,,D 5 1, a comparison of the Love waves velocities using either the exact calculations (eqs. (7)) or the effective medium approximation (EMA) is given in fig. 3 as a function of k,, L, where L is the thickness of the superlattice. The curves 1 (respectively 2) correspond to a superlattice with Dy (respectively Y) layers at both ends and containing 5.5 periods, i.e. N = 5. We also obtain results very close to curves 1 (respectively 2) when the medium at the surface is still Dy (respectively Y) but the film at the interface is Y (respectively Dy). In fig. 3, at k,, L = 1 for example, the velocity difference between curves 1 and 2 is of the order of 50 ms-‘. The EMA results (dashed curves in fig. 3) are obtained by replacing the superlattice with an homogeneous material whose parameters p”, Ct6 and l/C& are given by the arithmetic averages of the corresponding quantities in the superlattice; these curves are of course independent of the number of periods N and the nature of the utmost layers in the superlattice. The exact results become closer to the EMA curves when N increases; this also shows that for a given thickness L of the superlattice one obtains different Love waves velocities by varying simultaneously N and D.
366
E. Khourdifi,
B. Djafari Rouhanr
OS_ 1
/ Love and shear horizontal
1~ 5
I
4
6
mterface
waues
kfA
Fig. 3. Velocities of Love waves as a function of k ,, L, where L is the thickness of the superlattice. The curves 1 (respectively 2) correspond to a superlattice with N = 5 and Dy (respectively Y) layers at its ends. The substrate is the same as in fig. 2. The dashed curves are the result of the effective medium approximation.
In conclusion we have shown for the first time the possibility of shear horizontal interface waves at the boundary between a superlattice and a substrate. It is worth to mention that such a mode does not exist if the substrate is one of the two materials out of which the superlattice is fabricated. For a finite superlattice we have discussed the dispersion of the Love waves as a function of k ,,D and the number of periods N. These results are dependent upon the nature of the interface and surface layers in the superlattice.
References PI See for example:
J. Warlock, in: Proc. 2nd Intern. Conf. on Phonon Physics (World Scientific, Singapore, 1985) p. 506; M.V. Klein, IEEE J. Quantum Electron. 22 (1986) 1760; B. Djafari Rouhani, Lecture Notes in Physics, Vol. 285 (Springer, Berlin, 1987) p, 148. PI A. Kueny, M. Grimsditch, K. Miyano, I. Banerjee. C. Falco and I.K. Schuller, Phys. Rev. Letters 48 (1982) 166. R. Vacher, J. Kervarec and A. Regreny, Phya. Rev. B [31 J. Sapriel, J.C. Michel, J.C. Toledano. 28 (1983) 2007. A. Kueny, C.M.Falco and 1.K. Schuller. 141 M.R. Kan. C.S.L. Chun, G. Felcher. M. Grimsditch. Phys. Rev. B 27 (1983) 718. f51 R. Danner, R.P. Huebener, C.S.L. Chun. M. Grimsditch and I.K. Schuller, Phys. Rev. B 33 (1986) 3696.
E. Khourdifi, 8. Djafari Rouhani / Low and shear honzontal interface waues
367
[6] P. Bisanti, M.B. Brodsky, G.P. Felcher, M. Grimsditch and L. Sill, Phys. Rev. B 35 (1987) 7813. [7] J.A. Bell, W.R. Bennett, R. Zanoni, G.I. Stegeman, CM. Falco and F. Nizzoli, Phys. Rev. B 35 (1987) 4127. [S] P. Baumgart, B. Hillebrands, R. Mock, G. Giintherodt, A. Boufelfel and CM. Falco, Phys. Rev. B 34 (1986) 9004. 191 J.A. Bell, W.R. Bennett, R. Zanoni, G.I. Stegeman, CM. Falco and CT. Seaton, Solid State Commun. 64 (1987) 1339. [IO] J.A. Bell, R.J. Zanoni, CT. Seaton, G.I. Stegeman, W.R. Bennett and CM. Falco, Appl. Phys. Letters 9 (1987) 652. [ll] B. Djafari Rouhani, L. Dobrzynski, 0. Hardouin Duparc, R.E. Camely and A.A. Maradudin, Phys. Rev. B 28 (1983) 1711; R.E. Camley, B. Djafari Rouhani, L. Dobrzynski and A.A. Maradudin, Phys. Rev. B 27 (1983) 7318. (121 A. Nougaoui and B. Djafari Rouhani, Surface Sci. IS5 (1987) 125. [13] L. Dobrzynski, Surface Sci. 182 (1987) 362. [14] M.B. Salamon. S. Sinha, J.J. Rhyne, J.E. Cunningham R.W. Erwin, J. Borchers and C.P. Flynn, Phys. Rev. Letters 56 (1986) 259.